Matter, Energy, and Temperature

To understand the world of fundamental particles, we must first become familiar with the extreme scales on which nature operates – and with the way size, energy, and temperature are interconnected. That is the aim of this chapter.

Orders of magnitude and scales

An adult human is about 1.7 m tall – in scientific notation: the order of magnitude of $10^0$ m. That notation may sound abstract, but it is indispensable in physics. The universe spans a range of more than 44 orders of magnitude: from the smallest particles we know ($< 10^{-18}$ m) to the observable universe ($\sim 10^{26}$ m). A single step in order of magnitude means a factor of ten.

To make such extremes manageable, physics uses adapted units:

• Ångström (Å) – $10^{-10}$ m: the typical size of an atom.
• Fermi (fm) – $10^{-15}$ m: the scale of an atomic nucleus.
• Light-year – $9.46 \times 10^{15}$ m: the distance light travels in one year.

Within our solar system, Earth itself is $\sim 10^7$ m across, the Sun $\sim 10^9$ m, and Earth's orbit around the Sun $\sim 10^{11}$ m. Our galaxy measures $\sim 10^{21}$ m. Galaxies cluster into groups ($\sim 10^{23}$ m) and superclusters ($\sim 10^{24}$ m); the observable universe extends to $\sim 10^{26}$ m. Each step in that sequence represents at least a factor of ten to a hundred in distance.

At the atomic scale the ratio is even more extreme. An atom ($10^{-10}$ m) relates to its nucleus ($10^{-14}$ m) as a football stadium relates to a marble. Electrons and quarks are smaller than $10^{-18}$ m – a ratio of 1 to 10,000 relative to the nucleus. An atom is therefore overwhelmingly empty space.

Energy at subatomic scales

On the human scale, we measure energy in joules. For subatomic particles, however, the joule is enormously inconvenient: an electron accelerated through a 1 V battery gains only $1.6 \times 10^{-19}$ J. Even in the world's most powerful accelerators, particles reach 'only' $\sim 10^{-5}$ J – comparable to the kinetic energy of a fly.

Particle physics therefore uses its own unit: the electronvolt (eV):

$1\,\mathrm{eV} = 1.6 \times 10^{-19}\,\mathrm{J}$

For higher energies, familiar prefixes apply: keV ($10^3$ eV), MeV ($10^6$ eV), GeV ($10^9$ eV), and TeV ($10^{12}$ eV).

The electronvolt also connects energy to mass, through Einstein's $E = mc^2$. Because the speed of light $c$ is a fixed constant, every mass has a direct energy equivalent. A proton thus has a mass of 938.3 MeV/$c^2$ and an electron 511 keV/$c^2$. In practice, the $c^2$ is often dropped and physicists simply say 'a proton of 938 MeV'.

Temperature

Temperature is a macroscopic concept – not a property of a single particle, but a statistical average over a large ensemble of particles. The warmer a system, the greater the average kinetic energy with which its particles move and collide. This coupling is established by the Boltzmann constant ($k_B \approx 8.6 \times 10^{-5}$ eV/K): at room temperature ($\sim 300$ K), the average kinetic energy per particle is about 0.025 eV – the thermal energy.

That number has direct physical meaning. When the thermal energy of colliding particles becomes comparable to a binding energy, a collision is energetic enough to break that bond. The binding energy of the outermost electron in a neutral atom is typically 1–10 eV, corresponding to temperatures of $\sim 10^4$ K. Below that threshold, atoms are stable; above it, they are ionised as collisions become energetic enough to knock electrons free.

The same reasoning applies at higher scales. In the core of the Sun ($\sim 10^7$ K), the thermal energy is so large that no bound atom can survive. At $10^{10}$ K ($\sim 1$ MeV), collisions are so violent that new particles can be created: two photons then carry enough energy to materialise an electron–positron pair. Above $\sim 2$ GeV, protons and antiprotons can spontaneously emerge from pure energy.

Energy, wavelength, and 'seeing' structure

How do we look at something? Light falls on it, reflects, and reaches our eyes. But visible light has a wavelength of 400–700 nm ($\sim 10^{-7}$ m) – a thousand times larger than an atom. To 'see' something, the wavelength of the radiation used must be smaller than the object being observed: the wavelength determines the resolving power (the resolution) of the observation. Atom-scale structure requires X-rays ($\sim 10^{-10}$ m, 10–100 keV). For nuclei and quarks, even shorter wavelengths – and therefore higher energies – are required.

Quantum mechanics tells us that the energy and wavelength of radiation are directly linked:

$E = \frac{hc}{\lambda} \approx \frac{10^{-6}\,\mathrm{eV}\cdot\mathrm{m}}{\lambda}$

This is the central principle behind particle accelerators: the higher the energy, the smaller the structure we can probe.

This closes the triangle. The previous section linked temperature to energy ($E \sim k_BT$); this relation links energy to wavelength ($E = hc/\lambda$). A system at temperature $T$ emits radiation with a characteristic wavelength $\lambda \sim hc\,/\,k_BT$. The three quantities – energy, wavelength, and temperature – are interchangeable.

The interactive explorer below shows seven characteristic energy scales – from the cosmic microwave background to the LHC frontier. Select a scale to read off the corresponding wavelength and temperature.

Energy–Wavelength–Temperature Scale Explorer

Select an energy scale. The probe peak shows which physical structures can be resolved at that energy – read off the corresponding wavelength and temperature in the axis label.

The table below brings those scales together:

Energy	Wavelength	Temperature	Scale
0.3 meV	~4 mm	3 K	Cosmic microwave background
25 meV	~50 µm	300 K	Room temperature
1 eV	~1 µm	$10^4$ K	Atomic shell
1 keV	~1 nm	$10^7$ K	Solar core
1 MeV	~1 pm	$10^{10}$ K	Particle creation (electron)
1 GeV	~1 fm	$10^{13}$ K	Particle creation (proton)
1 TeV	~$10^{-18}$ m	$10^{16}$ K	LHC frontier

When cosmologists speak of 'the universe at a temperature of $10^{10}$ K', they mean precisely this: the average energy per particle at that moment was ~1 MeV, with a corresponding wavelength of ~1 pm. At that energy scale, processes such as the creation of electron–positron pairs dominated – and that determined what the universe looked like in that phase. The table therefore translates not only laboratory physics into cosmology, but also the reverse: each step in temperature reveals a different layer of nature.